A couple weeks ago, I posted a puzzle here that was "too easy" according to some; of course, it came after a stretch of tough Thomasyu, Nurikabe, etc., but still, not all puzzles have to be impossible in my opinion to at least raise a discussion of whether a new idea has some promise. In the case of that Digital Pent By Number, I think the discussion led to a better potential variant on that design which would have more challenge and I'll try to make such a puzzle soon. This week's puzzle deserves a different description than "too easy". Similar to some recent ideas posted by onigame, this puzzle is too hard (in my opinion at least) but certainly not impossible.

When writing a lot of puzzles, there are several ways to get some that are unexpectedly hard. Using a computer to strictly tell you there is 1 answer (without checking a solution path yourself) is a good way to get some really bad outliers. I'll almost never encounter this problem in my designs, but I encounter it all the time with work by other designers. Trying to make very minimal designs, or artistic designs with very specific constraints, is another way to get a puzzle that is too hard. In some designs, you might have a hard time just filling a valid grid with digits and so considerations of difficulty don't come into mind until you reach an answer and experiment with how to clue a puzzle to reach it. Finally, you can run into non-uniqueness at the end of a construction, requiring some digit tweaking which can compromise intended work-ins and possibly making the puzzle much easier or much harder than expected. The puzzle here is primarily a result of the last case.

In my forthcoming book of hand-crafted calcudoku puzzles there will be an 8x8 grid called "A Grate Puzzle" (yes, I'm shameless when it comes to titling sometimes). Anyway, the puzzle below was a first attempt to use the same theme (shape of cages plus geometric isolation of each of the four operations to one quadrant) but simply is "too hard to be published" in my opinion and got a full rewrite for the book. However, one thing I realize is that I have very little sense of what is or is not hard for my (blog) audience. So, to those who would take up this gauntlet, here is a puzzle that I could eventually solve in around twenty minutes (I wasn't timing but this seems about right), that has 2 or 3 solid chunks of progress that follow much longer periods of staring.

Note: As always, although it is brought up each and every time by solvers of those other computer-generated puzzles, multi-cell subtraction and division starts with the largest value with all digits subtracted/divided from it. A cage with 1, 4, 2 and - is 4-2-1=1 or the same digits with / is 4/2/1=2.

A tough and fun-to-solve puzzle, and a very nice layout and design. Solution and a few spoilers below:

A good place to start was with the fives, sevens, and eights in the bottom half of the puzzle. Also, you can determine the product of the numerators in the division section only by taking into account the clues in the multiplication and division sections. You can do a similar thing in the top section, and it helps a bit to know what the minuends (had to look that word up) are in the subtraction section.

Another trick (which is not needed to solve but is related to what you are commenting about links between sections) is that: By Law of Leftovers, the 16 digits in the upper-left/lower-right are the same as are the opposite sets of corners. So, while you may not have the full location of products, knowing those 16 digits will let you break in much quicker to the sums as you prove there are no 1's, only two 8's, only two 7's, ...